A060595 Number of tilings of the 3-dimensional zonotope constructed from D vectors.

1, 2, 10, 148, 7686, 1681104, 1881850464, 13227777493060

Offset: 3

Author

Matthieu Latapy (latapy(AT)liafa.jussieu.fr), Apr 12 2001

Keywords

• nonn
• nice

Comments

The zonotope Z(D,d) is the projection of the D-dimensional hypercube onto the d-dimensional space and the tiles are the projections of the d-dimensional faces of the hypercube. Here d=3 and D varies.

Also the number of signotopes of rank 4, i.e., mappings X:{{1..n} choose 4}->{+,-} such that for any four indices a < b < c < d < e, the sequence X(a,b,c,d), X(a,b,c,e), X(a,b,d,e), X(a,c,d,e), X(b,c,d,e), changes its sign at most once (see Felsner-Weil reference). - Manfred Scheucher, Sep 13 2021

Examples

			Z(3,3) is simply a cube and the only possible tile is Z(3,3) itself, therefore the first term of the series is 1. It is well known that there are always two d-tilings of Z(d+1,d), therefore the second term is 2. More examples are available on my web page.
		

References

• A. Bjorner, M. Las Vergnas, B. Sturmfels, N. White and G.M. Ziegler, Oriented Matroids, Encyclopedia of Mathematics 46, Second Edition, Cambridge University Press, 1999
• V. Reiner, The generalized Baues problem, in New Perspectives in Algebraic Combinatorics (Berkeley, CA, 1996-1997), 293-336, Math. Sci. Res. Inst. Publ., 38, Cambridge Univ. Press, Cambridge, 1999.

Links

• Helena Bergold, Stefan Felsner, and Manfred Scheucher, Extendability of higher dimensional signotopes, Proc. 38th Eur. Wksp. Comp. Geom. (EuroCG), 2022. See also arXiv:2303.04079 [math.CO], 2023.
• N. Destainville, R. Mosseri and F. Bailly, Fixed-boundary octagonal random tilings: a combinatorial approach, arXiv:cond-mat/0004145 [cond-mat.stat-mech], 2000.
• N. Destainville, R. Mosseri and F. Bailly, Fixed-boundary octagonal random tilings: a combinatorial approach, Journal of Statistical Physics, 102 (2001), no. 1-2, 147-190.
• S. Felsner and H. Weil, Sweeps, arrangements and signotopes, Discrete Applied Mathematics, Volume 109, Issues 1-2, 2001, Pages 67-94.
• M. Latapy, Generalized Integer Partitions, Tilings of Zonotopes and Lattices, arXiv:math/0008022 [math.CO], 2000.
• J. A. Olarte and F. Santos, Hypersimplicial subdivisions, arXiv:1906.05764 [math.CO], 2019.
• Manfred Scheucher, C program for enumeration
• G. M. Ziegler, Higher Bruhat Orders and Cyclic Hyperplane Arrangements, Topology, Volume 32, 1993.

Crossrefs

Cf. A006245 (two-dimensional tilings), A060596-A060602.

Column k=3 of A060637.

Formula

Asymptotics: a(n) = 2^(Theta(n^3)). This is Bachmann-Landau notation, that is, there are constants n_0, c, and d, such that for every n >= n_0 the inequality 2^{c n^3} <= a(n) <= 2^{d n^3} is satisfied. - Manfred Scheucher, Sep 22 2021

Extensions

a(8)-a(9) from Manfred Scheucher, Sep 13 2021

Edited by Manfred Scheucher, Mar 08 2022

a(10) from Manfred Scheucher, Jul 17 2023